代做MATH502 Assignment 1 S2 2024代做Java语言

MATH502 Assignment 1 S2 2024

Due Date: Friday 16th August 2024, 5:00pm

Instructions

•  There are a total of 9 questions, consisting of a total of 100 marks.

•  The assignment should be submitted as a  single PDF document through Canvas by the due date. The assignment can be either handwritten or typed on A4 size sheets, and must be submitted as a pdf. If you hand-write, a recommended app to convert to pdf is ‘camscanner’, but any method that gives a reasonably sized pdf is fine.

•  Remember to include your name and ID number clearly in what you submit.

•  Remember to upload the signed and dated cover sheet.

•  All working must be given for all questions, even if the question does not specifically ask for it. If a question does not ask for a particular method of solution, any method we have covered in class is fine. Unless specifically mentioned, questions must be worked out by hand, not on a computer.

•  This is an individual assignment, although you are encouraged to work together with others to discuss questions and come up with general strategies for solutions, what you hand in must be your own work.

•  Late assignments, without an approved extension, will be subject to a deduction of 5% (one grade e.g. from C+ to C) of the total mark available for each 24-hour period, or part thereof, up to a maximum of five calendar days. Assignments over five days late will not normally be accepted or marked and students will receive a DNC (Did Not Complete) for that assessment. You may find more information from the URL link: https://student.aut.ac.nz/study/study-help/assessments-and-exams

Question 1 (10 marks) Use truth tables to show the following:

(a)  whether →p Λ → (p → →q) is a tautology, a contradiction or neither.

(b)  whether ((p → q) Λ (p → r)) → (p → (q Λ r)) is a tautology, a contradiction or neither.

(c)  → (→p Λ q)  and q → p are logically equivalent.

(d)  ((p → r) Λ (q → r)) Λ →r  and → ((p V q) V r)  are logically equivalent.

Question 2 (10 marks) Use laws of logic to show the following equivalences  (clearly indicate  which law you use in  each step):

(a)  → (→p Λ q) ≡ (q → p) .

(b)  ((p → r) Λ (q → r)) Λ →r ≡ → ((p V q) V r)

Question 3 (10 marks) Using Euclid’s  algorithm, determine if the following rational numbers are in re- duced form. If not, write down  their reduced forms. Show  your working.

(a) 76/180

(b) 53/223

Question 4 (10 marks) Prove the following statements.

(a)  For  all n ∈ Z, n3 + 3n2 + 2n ≡ 0  (mod 3).

(b)  For  all n ∈ Z, n3  + 3n + 2 is  even.

Question 5 (20 marks) Prove  or disprove  the following as  directed:

(a)  Prove the following  by direct proof:

Let a ∈ Z,  if a ≡ 1 (mod 7),  then a2  ≡ 1  (mod 7) .a2  ≡ 1 (mod 7)

(b)  Prove the following by contraposition  (using  the  contrapositive): For all n ∈ Z, if n2  is  even,  then n is  even.

(c)  Disprove the following by finding counterexamples:

• For  all integers a,b,  if a + b is  even  then  both  a and b are  even.

• Let A, B and C be sets.  If A × C = B × C,  then A = B .

(d)  Prove  by  induction  that 2n + 1 ≤ 2n  for  all  integers n ≥ 5.

(e)  Prove  the following  by  induction:

n2  < 2n for all integers n ≥ 5. Make use of the result in the previous part.

Question 6 (10 marks) Let A = {1, 4, 5, 6}, B  = {2, 3, 4, 5},  C  = {1, 2, 3}. Show the following sets by enumeration:

(a)  {x | x ∉ A,x ∈ B  and x|24}

(b)  {X | X ⊆ A and |X| = 2}

(c)  {X | X ⊆ A and 4 ∈ X}

(d)  {X | X ⊆ B  and X ∩ A ∅}

(e)  The power set of C.

Question 7 (10 marks) Let U,A,B, C  be  as indicated in the  Venn  diagram  below:

Shade the areas  of following sets in  Venn  diagrams.

(a)  (A ∩ B) ∩ C

(b)  A ∩ (B ∩ C)

(c)  A ∪ C ∩ B

(d)  A ∪ (B ∪ C))

Question 8 (10 marks) Diet Survey

In a survey of 99 individual students regarding diet,  the following data were  obtained:

• 57 eat apples.

• 45  eat broccoli.

• 56 eat cheese.

• 22  eat  apples  and  broccoli.

• 25  eat  apples  and  cheese.

• 27 eat  broccoli  and  cheese.

• 5  do  not  eat  any  of these  three foods.

(a)  Represent this problem in a Venn diagram, and shade the set of students who eat only one of the  three foods.

(b)  How many students eat only one of the three foods?  (Hint: you may want to first figure out how many students  eat all three foods.)  Show  your working.

Question 9 (10 marks) Use set operation laws to show the following set equality. Clearly indicate which law you use in  each step.

((A ∪ C) ∩ (B ∪ C)) ∩ C = (A ∩ B) ∩ C